1e:I[80107,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1f:I[76763,["/_next/static/chunks/235ad55241b6443f.js","/_next/static/chunks/8337ab1f037b6654.js","/_next/static/chunks/c9e933b5fd1a0afc.js","/_next/static/chunks/848a330faa2461ac.js","/_next/static/chunks/9f2942e368a262d2.js","/_next/static/chunks/92f9e11244990470.js","/_next/static/chunks/a8b867fbb1dfca1a.js","/_next/static/chunks/1100f0390a901fa7.js","/_next/static/chunks/bbe6bd9d3c7b0355.js","/_next/static/chunks/a82195c3e5de7dd4.js"],"default"] 1d:T43d,{"@context":"https://schema.org","@graph":[{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://www.divinejee.com"},{"@type":"ListItem","position":4,"name":"Let a curve yfx x 0 fty pass through the points P1 frac 32 a","item":"https://www.divinejee.com/question/let-a-curve-yfx-x-0-fty-pass-through-the-points-p1-frac-32-and-qa-frac-12-if-the-tangent-at-any"}]},{"@type":"QAPage","mainEntity":{"@type":"Question","name":"Question Let a curve yfx x 0 fty pass through the points P1 frac 32 a on Topic","text":"

Let a curve $y=f(x), x \\in(0, \\infty)$ pass through the points $P\\left(1, \\frac{3}{2}\\right)$ and $Q\\left(a, \\frac{1}{2}\\right)$. If the tangent at...","answerCount":1,"upvoteCount":0,"datePublished":"2026-01-16","author":{"@type":"Organization","name":"DivineJEE"},"acceptedAnswer":{"@type":"Answer","text":"Check the detailed solution on the page.","upvoteCount":0,"url":"https://www.divinejee.com/question/let-a-curve-yfx-x-0-fty-pass-through-the-points-p1-frac-32-and-qa-frac-12-if-the-tangent-at-any"}}}]}20:T453,Equation of tangent at $R(b, f(b))$

$$ y-f(b)=f^{\prime}(b)(x-b) $$

which passes through $S(0, c)$

$$ \begin{aligned} & \therefore c-f(b)=f^{\prime}(b)(0-b) \\\\ & b f^{\prime}(b)-f(b)=-c \\\\ & \Rightarrow b f^{\prime}(b)-f(b)=\frac{-3}{b} (\because b c=3) \\\\ & \Rightarrow \frac{b f^{\prime}(b)-f(b)}{b^2}=\frac{-3}{b^3} \\\\ & \Rightarrow d\left(\frac{f(b)}{b}\right)=\frac{-3}{b^3} \\\\ & \Rightarrow \frac{f(b)}{b}=\frac{3}{2 b^2}+c \end{aligned} $$

which passes through $P\left(1, \frac{3}{2}\right)$

$$ \begin{aligned} & \Rightarrow \frac{3 / 2}{1}=\frac{3}{2}+c \\\\ & \Rightarrow c=0 \\\\ & \therefore f(b)=\frac{3}{2 b^2} \times b \\\\ & \Rightarrow f(b)=\frac{3}{2 b} \end{aligned} $$

$\because$ It passes through $Q\left(a, \frac{1}{2}\right)$

$$ \begin{aligned} & \therefore \frac{1}{2}=\frac{3}{2 a} \\\\ & \Rightarrow a=3 \\\\ & \therefore P \equiv\left(1, \frac{3}{2}\right) \text { and } Q \equiv\left(3, \frac{1}{2}\right)\\\\ & \therefore (P Q)^2=(3-1)^2+\left(\frac{1}{2}-\frac{3}{2}\right)^2=4+1=5 \end{aligned} $$