In the figure given above, PA and PB are tangents to the circle from point P. How many real values of $x$ exist such that the length of PA is $x$ and length of PB is $x^{2} + 1$ ?

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Solution

The correct option is A 0
Tangents from an external point to a circle are equal in length. So, the lengths of PA and PB are equal.
We know that $P A = x a n d P B = x^{2} + 1$
PA = PB
$x = x^{2} + 1$
$x^{2} - x + 1 = 0$
Now, this is a quadratic equation.
If we compare to the standard form $a x^{2} + b x + c = 0, a = 1, b = - 1 a n d c = 1.$
$b^{2} - 4 a c = (- 1)^{2} - 4 (1) (1) = 1 - 4 = - 3.$
Since $b^{2} - 4 a c < 0$ , the equation will have no real roots.
Hence, there are no real values of $x$ , that satisfy the given condition.

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In the figure given above, PA and PB are tangents to the circle from point P. How many real values of x exist such that the length of PA is x and length of PB is x2+1?

In the figure given above, PA and PB are tangents to the circle from point P. How many real values of $x$ exist such that the length of PA is $x$ and length of PB is $x^{2} + 1$ ?