Question

Assertion :Point $(5,5)$ lies inside the hyperbola $7x^2-2y^2+12xy-2x+14y-22=0$ Reason: Let $S=7x^2-2y^2+12xy-2x+14y-22$ and P$(5,5)$ then $S_1<0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Equation of given Hyperbola is $7x^2-2y^2+12xy-2x+14y-22=0$

Let $S=7x^2-2y^2+12xy-2x+14y-22$

At any given point $P(x_1,y_1)$ $\to S_1=7x_1^2-2y_1^2+12x_1{y}_1-2x_1+14y_1-22$

Now according to Properties of hyperbola if $S_1<0$ then point $P(x_1,y_1)$ lies outside the hyperbola.

if $S_1=0$ then point $P(x_1,y_1)$ lies on the hyperbola.

and if $S_1>0$ then point $P(x_1,y_1)$ lies inside the hyperbola.

Now for point $(5,5)$, $S_{(5,5)}=463......(>0)$ .......$eq.\left(1\right)$

So we can see that $(5,5)$ point gives a positive value of $S_1$,

$\because$ $S_{(5,5)}>0$, Hence point $(5,5)$ lies inside of the given hyperbola. So Assertion is correct.

Also from the $eq.\left(1\right)$ we can see that for point $(5,5)$, $S_{(5,5)}=463>0$ That's why the reason is incorrect.

So correct option is $C$.